Multivariable Calculus

Course Objective

At the end of this course students will be able to ...

1. differentiate functions of several variables (partial derivatives), find local extreme values and use these to graph functions;
   
2. parametrize curves and surfaces;
   
3. apply the implicit and inverse function theorem;
   
4. calculate and investigate multivariable Taylor polynomials of functions of several variables;
   
5. calculate multivariable integrals (2D and 3D integrals) using appropriately chosen methods, such as the substitution method, integration by parts and changing the order of integration;
   
6. investigate vector fields and line integrals;
   
7. work with differential k-forms;
   
8. formulate (the general) Stokes theorem and derive the classical integral theorems of Gauss, Green and Stokes;
   
9. write down the arguments involved in solving a calculus problem in a logically correct manner.

Course Content

This course deals with the calculus of functions of several variables.
In particular, we cover

• parametrized curves and arc length
  
• planes and lines
  
• functions of several variables and level sets
  
• partial derivatives, gradients and directional derivatives
  
• tangent planes and multivariable Taylor polynomials
  
• the multivariable chain rule
  
• the implicit and inverse function theorem
  
• optimization and optimization under constraints
  
• 2D integrals, order of integration
  
• 3D integrals, cylindrical and spherical coordinates
  
• changes of variables
  
• vector fields
  
• line integrals and surface integrals
  
• parametrized hyper-sufaces and manifolds
  
• differential k-forms
  
• (the general) Stokes theorem and the classical integral theorems of Gauss, Green and Stokes

Additional Information Teaching Methods

Method of Assessment

Weekly MyMathLab exercises (10%), one Midterm exams (35%) and a Final exam (55%). MML and both exams are mandatory.

The resit exam counts for 90%, with the 10% of the MyMathLab exercises still counting for the resit grade. There is no resit opportunity for the MyMathLab exercises.

Literature

• Calculus: A Complete Course, by Adams and Essex, 10th edition, Pearson, 2022. ISBN 978-0-13-573258-8

Additional Information Target Audience

Explanation Canvas

Course on differential forms. Available via Canvas.

Recommended background knowledge

Necessary background: Single Variable Calculus (XB_41007)